You cannot just multiply differentials like that: If x= u(s, t) and y= v(s, t) then where J(x,y;u,v) is the Jacobian determinant: .

In this case, u and v are r and , respectively, and so .

(In more advanced mathematics, differential geometry, we use that to define the "algebra of differentials" in such a way that it is anti-commutative- that is, that ab= -ba. From that so all squares are 0.

If then and if then .

Then . The terms we would get by multiplying dr and dr together or and together are 0 leaving which, since multiplication is anti-commutative, is the same as .)

October 7th 2010, 11:21 PM

rebghb

Dear HallsofIvy,

Thank you for commenting, I am aware of the Jacobian (stretching factor)... But never considered this case.
Regarding differential Geometry, can you recommend me a book (an intro) from calculus III to differential geometry? I would appreciate it...